\section{Quantitative Analysis of Insurer Operations}
\label{sec:QuantitativeAnalysisofInsurerOperations}

Insurers $NHI$, $B$, $PI$, $D$ and $E$ randomly select 308,000,000; 10,000,000; 1,000,000; 100,000; and 10,000 policyholders from the same population (See Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 1). Each insurer produces a Population Loss Ratio Estimate (PLRE) of the Population Loss Ratio. How accurate these PLREs are is measured by each insurer's portfolio size adjusted standard error. Insurers select large samples, so their PLRE distribution functions are all normally distributed, even if policyholder PLREs are not. I will show that transferring insurance risks to health care providers (Capitation) is flawed, by comparing the impact of portfolio size on PLRE variability, PLRE probabilities, and these five insurers' probabilities of earning Profits, incurring Operating Losses, or becoming Insolvent. I will also compare insurers' Surplus requirements and Maximum Sustainable Benefits for policyholders.

\subsection{Insurer Standard Errors by Portfolio Size}
\label{sec:InsurerStandardErrorsbyPortfolioSize}

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 1, shows insurer portfolio sizes in thousands (1,000s) of policyholders. Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 2, shows portfolio size adjusted standard errors, $\sigma_{e_{N}}$ = $\sigma_{e_{PI}}$ * $\frac{\sqrt{1,000,000}}{\sqrt{N}}$. $NHI$'s standard error, $\sigma_{e_{NHI}}$, is 0.00285 while $\sigma_{e_{E}}$ = 0.50000, ten times larger than $\sigma_{e_{PI}}$ (0.05000), and 175 times larger than $\sigma_{e_{NHI}}$. 

These insurers' normally distributed Population Loss Ratio Estimate Distribution Functions, are: $\Phi$(0.7500, 0.002849); $\Phi$(0.7500, 0.015811); $\Phi$(0.7500, 0.050000); $\Phi$(0.7500, 0.158114); and $\Phi$(0.7500, 0.500000), for $NHI$, $B$, $PI$, $D$ and $E$. %are normally distributed as described in Table~\ref{tab:CDF}.

Capitation advocates must have failed to note these profound differences in PLRE Distribution Functions because  \emph{all insurers larger than $PI$ have more probability below PLR + $\epsilon$} ($\epsilon > 0$) than $PI$, and \emph{all smaller insurers have more probability above PLR + $\epsilon$} than $PI$. When correctly analyzed, these subtle Distribution Functions result in dramatically different insurer operating results when insurance markets and health care (finance) systems are as efficient as possible, and policyholders are randomly selected.

% \begin{table}%[bbb] 
% \begin{center}
% \caption{Cumulative PLRE Distribution Functions by Portfolio Size} 
% \begin{tabular}{|crcc|} 
%  \hline
%   Insurer & \multicolumn{1}{c}{Size} &  Standard Error & Cumulative PLRE Distribution Function \\
% \hline
%   NHI & 308,000,000 	& 0.002849 & $\Phi$(0.7500, 0.002849) \\
%   B & 10,000,000 	& 0.015811 & $\Phi$(0.7500, 0.015811) \\
%   PI & 1,000,000  	& 0.050000 & $\Phi$(0.7500, 0.050000) \\
%   D & 100,000  		& 0.158114 & $\Phi$(0.7500, 0.158114) \\
%   E & 10,000  		& 0.500000 & $\Phi$(0.7500, 0.500000) \\
% \hline
% \end{tabular} \label{tab:CDF}
% %\caption{Cumulative Distribution Functions by Portfolio Size} 
% 
% \end{center}
% \end{table}


\subsection{Insurer Probabilities of Profits by Portfolio Size}
\label{sec:InsurerProbabilitiesofProfitsbyPortfolioSize}

Insurers use the 85\% of their premiums not allocated to operating expenses in Formula~\ref{eq:PremiumAllocationByCostsProspective} to pay policyholders' health expenses, converting all unused portions to profits. Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 3, shows all insurers have probability, $\Phi_N(0.7500)$ = 0.5000, of profits of at least 10\%, at PLREs at, or below, PLR (0.7500), because E[PLRE] = PLR for all insurers. Capitation advocates may not have gone any further than this because this is obviously the only PLRE value for which insurers' profit probabilities are identical. Small insurers have more probability in the tails of their distributions which is why they tend to have volatile operating results. Large insurers tend to have most of their probability close to the PLR, so their operating results tend to vary very little. 

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 4, shows insurers' probabilities of profits of at least 5\%, ($\Phi_N(0.8000)$), at PLREs below 0.8000. $NHI$ earns such profits with probability 1.0000, $B$ with probability 0.9992, and $PI$ with probability 0.8413. $D$ and $E$ have much lower probabilities of earning such profits, 0.6241 and 0.5398, respectively. However, $\Phi_{NHI}(0.8000)$) = 1.0000 is very misleading. $NHI$'s probability, $\Phi_{NHI}$(PLR + 3 * $\sigma_{e_{NHI}}$) = $\Phi_{NHI}(0.758547)$ = 0.9987. $NHI$ \emph{almost always} earns profits greater than 9.15\%, and since $\Phi_{B}(0.79743)$ = 0.9987, $B$ \emph{almost always} earns profits greater than 5.25\%! 

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 5, shows insurers' probabilities of profits greater than 0\% (Break Even), $\Phi_N(0.8500)$, at PLREs below 0.8500. $NHI$ and $B$ have probability 1.0000, $PI$'s probability is 0.9772, but $D$ and $E$ have much lower ``break-even`` probabilities, 0.7365 and 0.5793, respectively.

\subsection{Insurer Probabilities of Operating Losses by Portfolio Size}
\label{sec:InsurerProbabilitiesofOperatingLossesbyPortfolioSize}

$NHI$ and $B$ have probability 0.0000 of incurring operating losses, $PI$ has probability 0.0228, but $D$ and $E$ have much higher operating loss probabilities, 0.2635 (1.0000 - 0.7365) and 0.4207 (1.0000 - 0.5793), respectively. 

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 6, shows insurers' probabilities of incurring operating losses greater than 5\% ($1.0000 - \Phi_N(0.9000)$), at PLREs above 0.9000. $NHI$ and $B$ incur such operating losses with probability 0.0000, $PI$'s probability is 0.00135, but $D$ and $E$ have probabilities, 0.1714 and 0.3821, respectively. Insurer D is 127 times more likely, and Insurer E is 283 times more likely, to incur such operating losses, than $PI$. Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 7, shows insurers' probabilities of operating losses greater than 10\%, at PLREs above 0.9500. $\Phi_{NHI}$(0.9500) = $\Phi_{B}$(0.9500) = $\Phi_{PI}(0.9500)$ = 0.0000, while $\Phi_{D}(0.9500)$ = 0.1030 and $\Phi_{E}(0.9500)$ = 0.3446. $D$ can expect such high operating losses more than one year in ten, and $E$ more than one year in three.
